NASA Technical Reports Server (NTRS) 19770021980: Scaling of cross sections for K-electron capture by high-energy protons and alpha-particles from the multielectron atoms

General Disclaimer

One or more of the Following Statements may affect this Document

• This document has been reproduced from the best copy furnished by the
organizational source. It is being released in the interest of making available as
much information as possible.

• This document may contain data, which exceeds the sheet parameters. It was
furnished in this condition by the organizational source and is the best copy
available.

• This document may contain tone-on-tone or color graphs, charts and/or pictures,
which have been reproduced in black and white.

• This document is paginated as submitted by the original source.

• Portions of this document are not fully legible due to the historical nature of some
of the material. However, it is the best reproduction available from the original
submission.

Produced by the NASA Center for Aerospace Information (CASI)

I

f

X-620-77-151

PREPRINT

-^h X -

SCALING OF CROSS SECTIONS
FOR K-ELECTRON CAPTURE BY
HIGH-ENERGY PROTONS AND
ALPHA-PARTICLES FROM
THE MULTIELECTRON ATOMS

(NASA-TJl-71 J5b) SCALING CF CEOSS SECTIONS
fOu E-ELECTHCN CAEIUHE BY HIGH-ENEEGY
PBOTCNS AND A I E h A -P AHT IC LES EECM THE
MULIIELECTECK ATOMS (NASA) 21 p

HC AC3/MF A01 CSCL 20H GJ/72

K. OMIDVAR

JUNE 1977

N77-^:By24

Unc ids
39J2>4

GODDARD SPACE FLIGHT CENTER

GREENBELT, MARYLAND

SCALING OF CROSS SECTIONS FOR K-ELECTRON CAPTURE BY
HIGH-ENERGY PROTONS AND ALPHA-PARTICLES FROM THE MULTIELECTRON ATOMS

K. Omidyar

Stratosphere Physics and Chemistry Branch
Laboratory for Planetary Atmospheres
NASA/Goddard Space Flight Center
Greenbelt, Maryland 20771

ABSTRACT

Electron capture by protons from H, He, and the K-shell of Ar, and
alpha particles from He are considered. It is shown that when certain
function of the experimental cross sections is plotted versus the inverse
of the collision energy, at high energies the function falls on a straight
line. At lower energies the function concaves up or down, depending on
the charge of the projectile, the effective charge and the ionization
potential of the electron that is being captured. The plot can be used
to predict cross sections where experimental data are not available, and
as a guide in future experiments. High energy scaling formulas for K-
electron capture by low-charge projectiles are given.

1

I, INTRODUCTION

It is well known that an approximation developed by OppenheimerJ

2

and Brinkman and Kramers for electron capture by protons from the multi -
electron atoms results in cross sections that when plotted as functions
of the collision energy would give curves similar in shape to the experi-
mental curves, but are larger by as much as an order of magnitude. This
indicates that in the region of validity of the approximation the calcu-
lated cross section in so'^i’e respects has the correct functional form of
the actual cross section.

In this paper this approximation is used to calculate the K-electron
capture by protons and alpha-particles from the mul tielectron atoms with
the assumption of the hydrogenic wave functions with effective charges
for the K-shell electrons. A feature in the present calculation which
usually is absent in other calculations is to use the measured ionization
potential of the K-shell instead of the hydrogenic ionization potential
which is proportional to the square of the effective charge. This choice
to some extent compensates for the unphysical assumption that the target
potential is coulombic, and the neglect of the readjustment of the parent
ion when the capture takes place. Nikolaev^ has also introduced the
actual, instead of the hydrogenic, ionization potential. However, his
final results do not seem to be the same as those presented here.

Using the above prescription, the prior and post forms of the cross
section, and their asymptotic forms with respect to energy are given
analytically. Based on the asymptotic forms, a plot of a function of
the cross section versus the inverse of the energy is given where at

2

moderately high energies, when E„»M„Z , the experimental data falls

p P 6< J

on a straight line. Ep and Mp are the energy and mass of the projectile in
rydberg and electron mass units, and is the effective charge of the
target. For lower energies the lines concave up or concave down, depending
on the charge of the projectile, the effective charge and the ionization
potential of the K-shell.

It is shown that these features are satisfied using the experimental

data for electron capture by protons from H, He, and the K-shell of Ar,

and for electron capture by a-particles from He.

A scaling law that connects different charge exchange cross sections

to the p + H charge transfer is presented. Also, a formula with two

arbitrary parameters that are fitted to the experimental data for the K-

shell electron capture cross section is given that can be used for different

charge transfer reactions provided the projectile charge is not too large.

2

For energies Ep ^ 2 Mp Z^^^ the cress sections derived by this formula
are within a factor of 2 of the experimental data mentioned before.

For energies larger than 100 MeV/nucleon, the asymptotic form of
the capture cross section is governed by the second Born approximation^
which is different from the asymptotic foriii according to the first Born,
and the simple formula derived here is invalid in this energy region.

This energy region at present is beyond the interest of the experimenters.

3

II. FORMULATION

We consider capture of a K-shell electron by a proton or a struc-
tureless ion from a multielectron atom. The squared of the prior and
post forms of the exchange amplitude in the Oppenheimer-Brinkman-Kramers,

from here on called OBK, approximation and the assumption that the K-

5

electrons can be described by hydrogenic wave functions are given by

( 1 )

/ -r'’* /"_

o(. -

Jf

*13

J = ^^3

'ho OL

( 2 )

( 3 )

'S ^ ci ^ -ft - —

( 4 )

^ 9*0 .

In these equations n^ is the principal quantum number of the electron
after capture, and ^^3 reduced masses of the projectile +

electron and target nucleus + electrons, and and are the projec-
tile's charge and the effective charge of the K-shell electron in units
of the absolute value of the electronic charge e. m^ is the electron

mass, and a is the Bohr's radius. Ki and k., are the center of mass
0 ^

propagation vectors before and after collision. The magnitudes of k^

4

and k 2 are related through the conservation of energy by

^ ^ - J (X3)^ T(I,3J (5)

2 ^ I / •

where u-| and reduced masses of the system before and after

collision and I (2,3) and I (1,3) are the Ionization potentials of the
transferred electron before and after collision.

The cross section is then expressed through

-I

( 6 )

where I<i.k 2 = cos 0 with 0 the scattering angle.

It can be shown that most contribution to the cross section
comes from scattering angles 0 of the order m^/M, where M is the proton
mass. Then neglecting terms of the order m /M we obtain^

(7)

alU, Id- ]

( 8 )

In the above v-j is the initial relative velocity of the colliding
particles, and v^ is the Bohr's velocity, b and c are defined such that
bM and cM are the incident particle and target nucleus masses, al is
the initial minus the final ionization potentials of the transferred
electron in rydberg.

5

o c

The actual ionization potential of the K-electron is less than

2

Ryd due to the repulsion of the electrons outside the K-shell when
a K-electron is being removed to infinity. Let this difference be 6.
Then

( 10 )

By subtracting (7) from (6) we obtain

( 11 )

2_2

To carry out the integration in (6) we introduce y = (2bc/(b+c)) g
Keeping in mind that the main contribution to the integral in (6) comes
from small values of O, by combining (7), (8), (1) and (6) we obtain
for the prior form

oO

** ''

f X ^ ^ f

= I- n

( 12 )

(13)

in the derivation of (12) we have also assumed that k^/k^^l. Since U 2
differs from by terms of the order m^/M, Eq. (5) shows that the above
assumption is justified when the incident energy is much larger than the
difference in the binding energies. The expression for o in (13) is a

6

factor of 2 larger than the expression for o in (6), since we have
assumed 2 electrons for the K-shell.

U j

O I

Evaluation of the integral in (12) results in

V 'y=«

(14)

where s is the incident velocity in units of the average orbital velocity
of the K-electrons. With the help of (15) and (10) it can be shown with-
out any difficulty that P > <5^ always. Then the series in (14) always
converges.

Similar to the derivation of (14), the following expressiofi can
be derived for the post-form of the cross section:

The series converges in all practical cases. By letting 6^ = 6/Zg^^-^o
in (14) and (16) , which amounts to assuming hydrogenic potential for the K-
shell, the prior and post forms of the cross section reduce to the familiar
form of the OBK cross section with the assumption of the hydrogenic
potential^

7

D C

Since we are dealing with high energy approximation, it is appropriate

2

to have the cross section in terms of the inverse powers of s , Let

2

us introduce x = 1/s , then (14) can be written in the following form:

L i'’ Zp J / lo i-y y/F

IL-H-ri ' /> / J (1!

Making use of the definition of p, the right hand side can be expanded

2

in powers of x. In this way we obtain to terms of the order x :

3 7 I*. ^

^ (>>■»♦.) I ^ / + a

X-s'Vl

^ X f o * >

/*» -7 5^
X

(19)

I

Similarly, for the post-form of the cross section we obtain up to x'^:

x = s-\<i

(20)

The two expressions have the same dependence with respect to x'^, but they are
different in their dependence on x.

Equation (17) can also be written in a form similar to (19) or (20):

/

f'-4rjV

”, z;

&

(21)

3

For the resonance H + H charge exchange collisions, where Al^ = 6^ = 0,
and Zp = i^ight hand sides of (19) through (21) become

1 + 4 X, and the three forms of the cross sections become the same.

The forms of Eqs. (19) - (21) suggest that if the left hand sides
of these equations are plotted versus x, for x <<1 a straight line
is obtained. In the next section we will show that this linearity is
satisfied by the experimental data.

Similarly, Eqs. (19) and (20) suggest that if we plot the left hand
sides of these equations versus x, for the resonance H + H charge
exchange collisions we should obtain a straight line, while for non-
resonance charge transfers the plot should be concave up or down,
depending on the values of Zp, and the K-shell ionization

potentials. In the next section this aspect of the theory will also be
tested against the experimental data.

II. RESULTS AND DISCUSSIONS

Assuming relationships (19) and (20) are followed experimentally,

2 7 2 5 2

for s >> 1 the quantity a ( 113 ^ )/Zp becomes a function of s

9

c^ly, and if this quantity is plotted versus s , all the experimental
points should fall on a single curve. To see to what extent this
scaling law is obeyed at low energies and how the scaling law is
approached as s increases, in Figure 1 the product Z^^^ c (Tra^ )
for low energy electron capture by protons from atomic hydrogen,
helium, and the K-shell of argon are plotted versus s .

9

As 1s seen in the figure the agreement among the three curves
improves as the energy increases. At the positions of the maxima for the
He and Ar curves, the ordinate of the Ar curve is about twice the
He curve, and about 4 times of the H curve. The corresponding cross
sections in this region are many orders of magnitudes different from
each other. For example at s = 0.4 corresponding to 10 keV protons on H
the measured cross section is 10"15 ^m^ (Ref. 6), at s^ = 0.42 corres-
ponding to 30 keV protons on He the measured cross section is 1.9 x 10“^®
cm^ (Ref. 7), and at s^ = 0.351 corresponding to 3 MeV protons on Ar K-
shell , the experimental cross section is 2.72 x 10“23 cm^ (Ref. 8).

Due to the large variation of differv-nt cross sections, the agreement
obtained in Figure 1 is impressive.

No scaling here is made for the dependence of the charge-exchange
cross section on Z . A low velocity calculation by Olson and Salop^^

H

indicate that for high the cross section increases approximately
linearly with respect to Z , while a Z dependence is predicted by

r f

n 12

Presnyakov and Ulantsev, The classical binary encounter method
2

also gives a Z dependence.

Using the measured cross sections for 4 different charge-exchange

processes, in Figure 2, the left hand sides of (19) or (20), except

2

for a numerical factor, are plotted versus x = 1/s . According to these
equations, for x << 1 the experimental points should fall on a straight
line. As x increases these points should concave up or down, depending
on the values of Zp, Z^^^ and the target's ionization potential.

As is seen the linearity for small x is satisfied to a good
degree, specially for p + H and p + He points. As x becomes larger.

10

o o

I

!

the data for p + H continue to fall on a straight line, as predicted
by (19) and (20), while the data for p + He concave down, and those
for He^^ + He concave up, in accord with the calculation as will be
explained below. There are not enough data for P + Ar for low x to
make a judgment. Four of the five experimental points for this case lie
on a straight line that passes, presumably accidentally, through the y-axis at
the common point that theoretically all charge transfer cross sections
should pass through.

Using the values of 1 ^, and the K-shell ionization potentials

for the 3 non-hydrogenic target cases, we find that the right hand sides
of (19) and (20) are given by

1 + 2.95 x - 0.122 x^ Prior

1 + 2.46 X - 0.122 x^ Post

1 + 5.05 X + 0.389 x^
1 + 4.56 X + 0.389 x^

1 + 2.16 X + 0.521 x^ Prior)

2 \ p + Ar

1 + 1.86 X + 0.521 x^ Post J

++

Then for P + and He + cases we find agreement with measurements
as far as convexity and concavity are concerned. For p + Ar case where not
enough data is available, the theory predicts that the data should concave up.

It is of interest to note that the curve obtained by using the hydro-
genic potential for the cross section, Eq, (21), always concaves up,
contrary to the experimental data.

Suppose a projectile of mass M , charge Z , and energy E captures

H r P

an electron from a target of effective charge Zg^|r. Then for x« 1, or

o c

Ep/MpZ^f 1, witti Ep and Mp in units defined before, the cross section

0 -, 7 (E ) according to (19) - (21) is related to the cross section

Zp.Zeff P

for electron captv;re oy protons from H at energy E^, o-j ^

where g is the number of electrons in the target K-shell, and M is

the proton's mass. The condition x«l makes this formula inapplicable

to many cases of interest. The formula can be used as an estimate of

the cross section when the data fall approximately on each other.

1 4

As ar example, from the graph of Gil body and Riding the p + H

2

cross section at 130 keV is about 0.048 Tra^ . Using this value and

the scaling (22), we find that the cross section for the He^^ + He

2 17

case at 1.50 MeV is 0.079 na^ . From the graph of Pivovar et al .

the cross section for on He is about 0.055, which is 30% smaller

than the value predicted by (22).

An alternative way of finding a general formula for the charge-

exchange cross section is to assume that the right-hand sides of (19)

and (20) arc a pciynominal in x, and to find the coefficients of the

polynominals by least square fitting to the data. This has been done

for a polynominal up to two terms. The corresponding cross section is

given by a. 6

/ . ,1 X Tt GLo ^ -

I

I

where for determination of and we have considered all the data

given in Figure 2 in the range 0<X£0.50,

The ratio of the experimental cross section to a (scaled) is

plotted in Figure 3. This ratio in the figure ranges between 0.60 to

2

1.80. We conclude that for energies E such that E > 2 M Z r.,

r r I 1

formula (23) gives the experimental cross sections within a factor of 2.

In Figures 4 through 7 the data points of Figure 2 for each process
are redrawn, and are compared with the prior and post forms of the OBK
cross sections as given by (14) through (16). In the case of the p + H
the prior and post forms are given by a single curve.

For the non-hydrogenic targets, the two forms come together at high
energies, but for the low energies, the prior form agrees always better
with the measurements. The fact that the post form is a bad approximation
can be seen physically as follows:

In the post form the interaction which causes the transition is
between the captured electron and its parent ion. The parent ion is
assumed to be a point charge in the post form while in reality this ion
has a structure different from a point charge. This difficulty does not
arise in the prior form.

Another difficulty with the post form is that when the projectile
is a multiply charged ion, there is a coulomb interaction between the
particles in the final state, while in the post form the relative motion
of these particles is described by a plane wave. However, it has been
shown by Bates and Boyd that for energies in the keV region or larger
the effect of the coulomb repulsion between the particles can be neglected.

13

From Figures 4~7 it can be seen that as the energy increases,

the discrepancy between the experimental and OBK cross sections decreases.

As an example, it can be determined from Fig. 4 that for the p + H case

the ratio of the OBK to the experimental cross section decreases from 5.9

at 50 keV energy to 4.2 at 130 keV. In the p + He case. Fig, 5, this

ratio changes from 4.3 at 200 keV to 2.2 at 10.5 MeV. This decrease in

3 19

the ratio has also been noted by Nikolaev and Hal pern and Law. Based
on Fig. 5 we can conclude that this ratio never reaches unity.

Nikolaev introduces an energy dependent empirical factor which is de-
fined as the ratio of the experimental to the OBK cross sections for elec-
tron capture from all shells of the multielectron atoms. This ratio is
obtained by fitting the OBK approximation to the experimental data. It
approaches zero for energies very small and very large, and reaches a
maximum of 0.455 for v/(/ v^) % 1.62. Despite its usefulness, the

main difficulty with this factor is that in the energy region of interest
the ratio of the experimental to the OBK cross sections increases as
the energy increases, but this factor decreases with increasing energy.

In the present paper we have considered protons and alpha-particles
as projectiles. It is also of interest to consider the heavier bare nuclei.
The present scaling is not applicable to the heavier nuclei due to the fact
that for these nuclei electrons are captured predominantly in many excited
states, while the present treatment considers capture into a single state,
namely the ground state,

The quantum number ni for which the capture cross section is maximum
can be obtained by minimizing the product of the left hand side of (21)
and ni"3/5 t^^th respect to n-j . In this way we find that the cross section

14

becomes maximum for a value of such that

'A

Zp

z

(? 4 )

Then the value of for which the cross section becomes maximum is

2

directly proportional to Zp. For s >> 1 we obtain

no

/

a. 3?

5 %>/

(25)

+26

As an example, for a projectile of 10 MeV/nucleon energy corres-

ponding to v-|/Vq ^ 20 incident on atomic hydrogen the maximum cross section
occurs at n^ = 3. Capture cross section into the ground state will not
dominate until the energy is well above 100 MeV/nucleon.

IV. CONCLUSION

It is shown that the cross section according to the OBK approximation
represent correctly some aspect of the functional dependence of the available
experimental cross sections on the collision energy. A plot of a function of
the available experimental cross sections can be used to predict cross sections
where experimental data are not available, and as a guide for future experi-
ments. Some scaling formulas are given.

15

REFERENCES

1. J. R. Oppenheimer, Phys. Rev, 349 (1928).

2. H. C. Brinkman and H. A. Kramers, Proc. K. Ned. Akad, Wet,

973, (1930).

3. V. S. Nikolaev, Zh. Eksp. Teor. Fiz. 1263 (1966) [Sov‘. Phys, -
OETP 24, 847 (1967)].

4. P. J. Kramer, Phys. Rev. A 2125 (1972), and the references thereof.

5. J. D. Jackson and H. Schiff, Phys. Rev. 8^, 359 (1953).

6. H. Schiff, Can. J. Phys. 393 (1954).

7. R. F. Stebbings, A. C. H. Smith, and W. L. Fite, Proc. Roy. Soc. A
268 , 527 (1962).

8. S. K. Allison, Rev. Mod. Phys. 1137 (1958).

9. J. R. Macdonald, C. L. Cocke, and W. W. Eidson, Phys. Rev. Lett.

648 (1974).

10. R. E. Olson and A. Salop, Phys. Rev. A 14 , 579 (1976).

11. L. P. Presnyakov and A. D. Ulantsev, Kvant. Elektron 1, 2377 (1974)
[Sov. J. Quantum Electron 4, 1320 (1975)].

12. J. D. Garcia, E. Gerjuoy, and J. E. Welker, Phys. Rev. 165 , 72 (1968).

13. W. L. Fite, R. F. Stebbings, D. 6. Hummer, and R. T. Brackman, Phys.
Rev. 119, 663 (1960).

14. H. B. Gilbody and G. Ryding, Proc. Roy. Soc. A 291 , 438 (1966).

15. L. M. Welsh, K. H. Berkner, S. N. Kaplan, and R, V. Pyle, Phys, Rev.
158 , 85 (1967).

16. K. H. Berkner, S. N, Kaplan, G. A- Paulikas, and R. V. Pyle, Phys. Rev,
140, A729 (1967),

16

17. L. I. Pivoyar, M. T. Novikov, and V. M- Tubaev, Zh. Eksp. Teor.
Fiz. 1490 (1962) [Soy. PHys. - OETP 15, 1035 (1962)1.

18. D. R. Bates and A. H. Boyd, Proc. Phys. Soc. 1301 (1962).

19. A. M. Halpenn and J. Law, Phys. Rev. Lett. 31, 4 (1973).

17

FIGURE CAPTIONS

Figure T

Figure 2

Figure 3

7 2 2 2

. A plot of ^ versus v /(Z^^^v^) for electron

capture fcy p'^otons from H (Ref. 7), He (Ref. 8), and K-shell
of Ar (Ref. 9), The effective charges for He and K-shell
of Ar are assumed to be 1.69 and 17.4. Since there is one
K-shell electron for H instead of 2, for uniform scaling
the corresponding cross section has been multiplied by 2.

The positions on the abscissa where the cross section
according to the OBK approximation become maximum for He and
Ar targets are indicated by arrows. They can be compared to
the positions of the experimental maxima.

. A plot of ZpZg^^ [v/v^)''2 a(7ra^^)]"''/^ versus (Z^^^ v^)^/v^
for single electron capture in the following processes: i)

Proton on H (Fite et al. (Ref, 13) and Gil body et al . (Ref. 14)).
ii) Proton on (Allison et al . (Ref. 8), Welsh et al . (Ref. 15),
and Berkner et al . (Ref. 16)). iii) Proton on Ar (K-shell,

Mcdonald et al . (Ref. 9)). iv) Alpha-particles on H^ (Pivovar
et al. (Ref- 17)).

, A plot of the ratio of the experimental to the scaled, Eq. (23),
cross sections versus (v^ The ratio for each experi-

mental point is designated by the sign of the experimental point
used in Figure 2.

18

Figure 4

Figure 5

Figure 6

Figure 7

Comparison between the e.x;periiTiental data for H + H
charge exchange of Gil body and Ryding (Ref. 14) and the OBK
cross section. In this and the following three figures the
abscissas and ordinates are the same as in Figure 2,

Comparison between the experimental data for electron capture
by proton from and the prior and post forms of the OBK
cross section.

Comparison between the experimental data for electron capture
by protons from the K-shell of Ar (Macdonald et al. (Ref. 9))
and the prior and post forms of the OBK cross section.

Comparison between the experimental data for electron capture
++

by He from (Pivovar et al . (Ref. 17)) and the prior and
post forms of the OBK cross section.

r'

o c

o o o o o

lO'^COCN^-

c> d d d o

r

I

I ^

z

o

z

o

H

o

cc

a.

0) (b

X ^

l§

^ z

O O

2

Q.

(D ^
W 4 )

0) S ^

g -!2 5

= a > >-
> ®
> CD

<

+ X □

4 )

X

(N ~

t "

I !

i) tT3

3: g

+ 1

+

+ <

9 )

X

■i

■<

-(

+

-<

+

X

+

s

(Q31V

1.20

PROTON ON H

in

C>J

d

ooo

p

d

[ ua -7

o